The interpretations of multiwavelength observations and a full
understanding of nonlinear feedback effects in the interstellar gas
require simulations that include heating and cooling processes, and
the exchange between thermal phases. However, it is extremely
difficult to see how some of these effects should be treated. For
example, the short timescales of atomic and molecular cooling
processes relative to galaxy dynamical times presents one problem,
though, in fact, we need not time-resolve rapid cooling. Another
problem is the wide range of heating processes and scales associated
with energy input from stellar activity, including: UV photoheating,
winds from many different types of star, multiple supernova explosions
in young clusters, etc. Finally, adiabatic cooling in expanding flows
is probably one of the most important processes. In the turbulent
interstellar gas, expansion flows are likely to be found on many
length and time scales. Fortunately, we will usually only need to
resolve the effects of the scale on the interaction driven waves.

Very little work has been done in this area; there is almost no
literature to review. CS has begun some exploratory simulations with
very simple models for the thermal processes. In this subsection we
will very briefly describe some simulations of off-center
collisions. We expect it will be some time before there is any
standard or consensus method of modeling these processes, but the
example below provides an illustration of their importance. The
following description of the model is derived from
Struck (1995).

The simulation is based on a quite standard smooth particle
hydrodynamics (SPH) algorithm, which is essentially the same as that
used in
Struck-Marcell and Higdon
(1993).
19,000 particles were used
to represent the gas disk of the target galaxy and 3000 particles were
used to simulate the companion disk. The disk particles were
initialized on circular orbits in centrifugal balance with a fixed,
rigid gravitational potential (i.e., the halo) with a rotation curve
of the form
v
(r /
)1/n in the
target disk, where
v is
the rotation
velocity, and is the
scale length of the potential. With the
adopted value of n = 5, this rotation curve rises rapidly in the
inner regions, but is nearly flat at large radii. A simple softened
point-mass potential was used for the companion. These simulations did
not explicitly include either a stellar disk or a separate stellar
bulge component, and thus, represent late-type disk galaxies. The
simulations do include a calculation of local self-gravity between gas
particles on a scale comparable to the SPH smoothing length, as
described in
Struck-Marcell and Higdon
(1993).

The simulation presented below used a simple step function cooling
curve, with three steps. All gas elements are initialized to a single
temperature T0, assumed to lie in the range of about
5000-8000 K,
corresponding to the warm interstellar gas. At temperatures between
T0
and 70 T0 the cooling timescale
c is short
relative to other
timescales in this code, so it is assigned a value equal to a few
times the typical numerical timestep. This high cooling rate
represents the peak in the standard cooling curve due to hydrogen and
helium line emission in an optically thin gas. At temperatures above
70 T0 the cooling rate is decreased by a factor of
10. This decrease
is meant to approximate the combined effects of heating and cooling,
on the assumption that the hot gas is found near the sources of
heating. On the other hand, the hot gas frequently cools rapidly by
adiabatic expansion, so this part of the cooling function has a minor
effect on the model results. At temperatures below T0
the cooling rate
is decreased by a factor of 20 from the value immediately above
T0. Cooling is weaker at these temperatures in the
interstellar
medium. Moreover, another reason for using this decreased rate is to
inhibit the gas particles from cooling to negative temperatures, but
to do so without having to time-resolve the low temperature cooling.

When a gas particle is located in a dense or high pressure region
(cloud), so that its internal density exceeds a fixed critical value,
it is assumed that stars are formed. The particle is then heated by
increasing its temperature by a fixed multiple of T0
at each timestep for a finite duration, or until it reaches a maximum
Tmax. While it is
being heated it is not allowed to cool except by adiabatic
expansion. It is clear from the results that even these very simple
approximations are capable of representing phenomena that do not occur
in an isothermal gas.

Initially the gas is below the critical density for SF at all
locations. The gas density in the initial isothermal disk is also
below, but near, the threshold density for axisymmetric gravitational
instabilities, i.e. the value of the well-known Toomre Q parameter
indicates stability. Particles out of the plane of the disk are not
quite in centrifugal balance, and therefore settle into the disk at
the beginning of the simulation in the target disk. The companion is
so far below the threshold that it remains cold.

Figure 21a-c shows three orthogonal views
(x-y, x-z, y-z) of three
timesteps from this simulation (dimensionless length units are as in
Struck-Marcell and Higdon
1993).
In this figure the gas elements are
placed in three temperature bins, and are color-coded to indicate both
this, and which galaxy the particle originally belonged to. The
coolest elements, with temperatures less than 10 times the initial
value (i.e., < 50,000 K). Hot elements with temperatures greater than
300 times the initial and intermediate temperatures ("warm") are also
indicated by color.

Figure 21. Particle snapshots of an SPH
collisional simulation with heating and cooling as described in
Section 6.6. Each row contains three
orthogonal views - x-y, x-z,
y-z from left to right - at one time. The
three rows show the model at: a) thr time of impact, b) when the first
ring has expanded significantly into the disk and c) when the second
ring has almost reached the edge of the disk and a third ring has formed
at the center. Note the formation of strong spokes inside the second
ring in c). The color coding is as follows: For the target disk, red,
green and cyan represent the "cool", "warm", and "hot"
compnents. For the intruder galaxy, the same temperature sequence is
represented by the color yellow, magenta and blue respectively. Only
one in three SPH particles are show in these low resolution plot. See
Color Plate X at the back of this issue.

Figure 21a shows the disk at the time of
impact. The companion disk
is setup perpendicular to the primary disk in the y - z
plane. The orbit
of the companion center is in the x - z plane and it
impacts at a
significant angle. The strong shocks in the two disks are shown in the
first frame as a thin zone of hot, blue and cyan particles. The
effects of the shock are visible in the other two views, where it is
evident that the (yellow) disk does not simply punch through the
primary disk. The rotation directions of the two disks can also be
deduced from the small stochastic spirals in the unperturbed regions.

A significant part of the primary disk, and all of the companion
disk are disrupted by the impact, and much gas is splashed out in, a
bridge-plume. Nonetheless, most of the primary disk remains intact,
and the ring forms as in collisionless encounters. The second set of
frames (Figure 21b) shows a time when the first
ring has nearly
propagated through the disk. The ring is complete, but with azimuthal
density variations and significant warping. Green (warm) and blue
(hot) dots mark the sites of recent heating (i.e., star
formation). The preponderance of green relative to blue indicates that
the density is falling below the heating threshold. There is little
heating behind the ring, where strong cooling results from postshock
rarefaction.

The hot central regions and the blue-green are primarily the result
of infall from the bridge. It is also a result of azimuthal
compression in the off-center collision, as in
Toomre's (1978)
ring-to-spiral models. The fact that the infall is concentrated in
one narrow sector is surprising. However, it appears that most of this
infalling material originated in the half-disk where the rotation
velocity was opposite the orbital direction in the companion, or from
gas in the primary that it collided with (yellow and red appear quite
well mixed in the bridge).

Even more spectacular is the reformation of the companion disk,
which was almost totally disrupted in the collision. At this time the
disk is accreting from a stream of material that includes gas from the
companion that was orbiting in the same sense as companions orbit at
the time of impact. This material spirals into a smaller disk, so
there is much heating. The heating algorithm may not be all that
realistic here, but the accretion processes may in fact be responsible
for the enhanced activity in rings with attached companions.

The third set of frames (Figure 21c) is from a
much later time when
the companion has reached apoapse and begun to return. The ring at the
edge of the disk is now the second ring, and, though the disk is
warped, it is intrinsically noncircular. There are strong azimuthal
variations in the heating (star formation) around the ring, a
situation quite similar to that observed in the Cartwheel. A third
ring is forming in the central regions. The star formation in this
third ring is a result of the relatively high gas densities due to
infall there, and gas driven inwards by collisions with infalling
elements. This is not entirely realistic, because gas consumption has
not been included, and the heating algorithm only looks at the total
gas density, not what fraction of this gas is truly in a state
conducive to star formation.

This timestep also illustrates how the most distant companion (G3 of
Higdon 1993)
in the Cartwheel system might have managed to be the
intruder without having a very large relative velocity (despite its
size, the observed outer ring is the second ring in this model). The
scenario is especially compelling in the light of Higdon's recent
discovery (private communication) that the intergalactic HI plume
extends all the way to G3.

In this simulation, the most unstable wavelength in the
pre-collision primary disk, and the scale on which self-gravity is
computed, is quite small (0.1 in the simulation units). Thus it is not
surprising that there are no strong spokes in the wake of the first
ring (except for the "accretion arm"). Several strong spokes are
apparent behind the second ring (which becomes the outer ring in the
Cartwheel analogy). Because of their significant width in the third
dimension, the density in the spokes does not exceed the star
formation threshold. As noted above, this provides an alternative
explanation for the rarity of the spoke phenomenon - it may take two
cycles of wave compression and rarefaction to create them. The
simulation shows that there are other complications, such as infall
and the long-term effects of the "swing" in the off-center collision.

This simulation has not yet been analyzed in detail, nor have
parameter dependencies been studied with additional simulations. More
particles and spatial resolution in the multiphase gas disk would
certainly be desirable, as would the inclusion of gas consumption and
full selfgravity. Nonetheless, it illustrates some of the possible
effects of the thermal terms and is the first model we have seen of
gas exchange in a direct collision.